Geometric graph theory is a branch of mathematics that studies graphs in the context of geometry. It combines elements of graph theory, which is the study of graphs (composed of vertices connected by edges), with geometric concepts such as distances and shapes. The primary focus of geometric graph theory is on how graphs can be represented in a geometric space, typically the Euclidean plane or higher-dimensional spaces, while examining properties that arise from their geometric configurations.
Geometric graphs are a type of graph in which the vertices correspond to points in some geometric space, and the edges represent some geometric relationships between these points. The arrangement of the vertices in the plane (or in higher dimensions) usually relates to distances, angles, or other geometric properties. Key aspects of geometric graphs include: 1. **Vertex Representation**: The vertices are typically represented by points in a Euclidean space (commonly the 2D or 3D plane).
Boxicity is a mathematical concept related to graph theory. It refers to a particular way of representing a graph using boxes (or rectangles) in a Euclidean space. More specifically, the boxicity of a graph is defined as the minimum number of dimensions (d) such that the graph can be represented as the intersection of a family of axis-aligned boxes in \( \mathbb{R}^d \).
A contact graph is a type of graph used to represent relationships and interactions among entities, typically in the context of epidemiology, social networks, or communication networks. In a contact graph: - **Nodes (or Vertices):** Represent individual entities, which could be people, animals, or any other units of interest. - **Edges (or Links):** Represent the relationships or interactions between the nodes.
Convex embedding is a concept that arises in the fields of mathematics and computer science, particularly in the study of geometric properties and optimization problems. It generally refers to the process of transforming a given set of points or a geometric structure into a convex shape while preserving certain characteristics, such as distances or the arrangement of points.
In graph theory, the term "dimension" can refer to various concepts depending on the specific context in which it is used. Here are a few interpretations of dimension in relation to graphs: 1. **Graph Dimension**: In some contexts, particularly in the study of combinatorial or geometric properties of graphs, dimension may refer to the "Lemke-Howson" dimension or the "K-dimension". This is a way to measure how a graph can be embedded in a geometric space.
A Doubly Connected Edge List (DCEL) is a data structure used to represent a planar graph, especially in computational geometry. It provides a way to efficiently store and manipulate the relationships between edges, vertices, and faces of a planar graph. ### Components of a DCEL A DCEL typically consists of the following components: 1. **Edge**: Each edge in the DCEL contains: - A reference to its starting vertex.
The Flip Graph is a concept in combinatorial mathematics, specifically in the study of permutations and the arrangement of objects. It is a type of graph that represents the possible transformations (or "flips") of a given object, where nodes represent objects (or permutations) and edges represent allowable flips between them.
The term "slope number" can have different meanings depending on the context in which it is used, but it is not a standard term commonly found in mathematical literature.
A spatial network refers to a network that incorporates spatial relationships and geographic information into its structure, allowing for the representation and analysis of connected elements in a physical space. These networks can represent a variety of systems, including transportation networks (like roads, railways, and air routes), utility networks (such as water pipelines or electricity grids), social networks with geographic dimensions, and ecological networks that describe interactions among different species across habitats.
A theta graph is a type of graph used in the study of graph theory, particularly in the context of network flow problems and duality in optimization. Specifically, a theta graph is a form of representation that consists of two terminal vertices (often denoted as \( s \) and \( t \)), two or more paths connecting these vertices, and possibly some additional vertices that act as intermediate points along the paths.
Visibility Graph Analysis (VGA) is a method used primarily in the fields of spatial analysis, urban planning, landscape architecture, and other areas to assess spatial relationships and visibility within a given environment. It transforms physical spaces into a mathematical representation to analyze how different locations can be "seen" from one another, thus helping to understand visibility, accessibility, and spatial integration.
A Yao graph is a specific type of geometric graph used primarily in the field of computational geometry and computer science, particularly in the context of network design and algorithms. It was introduced by Andrew Yao in the 1980s. The Yao graph is constructed based on a set of points in a Euclidean space, usually in two or three dimensions.
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